12 problems

Quasiconvexity conjecture. The displayed function is quasiconvex at some matrix $A_0$. By homogeneity, quasiconvexity at $A_0$ implies quasiconvexity at $0$. This proposed strength…

Let $\mathcal{G}$ be a locally quasiconvex hyperbolic group. An immersion of finite directed height is an immersion $\psi:H\rightarrow F$ of graphs satisfying the finite directed-h…

Let $X$ be a compact $2$-complex with negative immersions. Write $\pi_1X$ for its fundamental group, and let local quasiconvexity and the finite-generation intersection property re…

Let $H<G$ be a hyperbolic subgroup of a hyperbolic group $G$. Suppose the Cannon–Thurston map … exists, and suppose that $H$ is not quasiconvex in $G$. Non-quasiconvex subgroup con…

Let $B_p\colon \mathbb{R}^{2\times 2}\to\mathbb{R}$ be the Burkholder integrand, defined by … where $\lvert A\rvert$ denotes the operator norm. Iwaniec's conjecture. The integrand…

The case $d=3$ conjecture. Any non-polyconvex weak extremal $f\in {\cal C}_3$ is an extreme ray of ${\cal C}_3$. Moreover, if $f\in {\cal C}_3$ is a non-polyconvex extreme ray of…

Assume that $H(p)=\varphi(|p|)$, where $\varphi:[0,\infty)\to\mathbb{R}$ satisfies (H7), and let … be the associated sequence, with $M_i=\varphi(s_{2i-1})$ and…

Let $f(\boldsymbol{\xi})=\boldsymbol{\xi}C\boldsymbol{\xi}^T$ be a quasiconvex quadratic form, where $\boldsymbol{\xi}\in\mathbb R^{3\times 3}$ and $C\in(\mathbb R^3)^4$ is a fourt…

Morrey's conjecture. Every rank-one convex function $\mathbf E:\mathbb R^{2\times2}\rightarrow\mathbb R$ is quasiconvex.

Relative quasiconvexity criterion. Then $H$ is relatively quasiconvex in $G$.

Let $d\geq 2$. For every subset $K\subset \mathbb{R}^d$, the main theorem asserts the existence of a connected set $\tilde\Gamma\subset\mathbb{R}^d$ containing $K$, with controlled…

Non-equivalence conjecture. If $m,n>1$ and $f$ is lower semicontinuous, then strong Morrey quasiconvexity and weak Morrey quasiconvexity are not equivalent.